Problem: $\dfrac{ -2d + e }{ 7 } = \dfrac{ d - f }{ -5 }$ Solve for $d$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -2d + e }{ {7} } = \dfrac{ d - f }{ -5 }$ ${7} \cdot \dfrac{ -2d + e }{ {7} } = {7} \cdot \dfrac{ d - f }{ -5 }$ $-2d + e = {7} \cdot \dfrac { d - f }{ -5 }$ Multiply both sides by the right denominator. $-2d + e = 7 \cdot \dfrac{ d - f }{ -{5} }$ $-{5} \cdot \left( -2d + e \right) = -{5} \cdot 7 \cdot \dfrac{ d - f }{ -{5} }$ $-{5} \cdot \left( -2d + e \right) = 7 \cdot \left( d - f \right)$ Distribute both sides $-{5} \cdot \left( -2d + e \right) = {7} \cdot \left( d - f \right)$ ${10}d - {5}e = {7}d - {7}f$ Combine $d$ terms on the left. ${10d} - 5e = {7d} - 7f$ ${3d} - 5e = -7f$ Move the $e$ term to the right. $3d - {5e} = -7f$ $3d = -7f + {5e}$ Isolate $d$ by dividing both sides by its coefficient. ${3}d = -7f + 5e$ $d = \dfrac{ -7f + 5e }{ {3} }$